Abstract
In the present paper and the companion paper (Berman, Kähler–Einstein metrics, canonical random point processes and birational geometry. arXiv:1307.3634, 2015) a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on complex algebraic varieties X is introduced by sampling “temperature deformed” determinantal point processes. The main new ingredient is a large deviation principle for Gibbs measures with singular Hamiltonians, which is proved in the present paper. As an application we show that the unique Kähler–Einstein metric with negative Ricci curvature on a canonically polarized algebraic manifold X emerges in the many particle limit of the canonical point processes on X. In the companion paper (Berman in 2015) the extension to algebraic varieties X with positive Kodaira dimension is given and a conjectural picture relating negative temperature states to the existence problem for Kähler–Einstein metrics with positive Ricci curvature is developed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alvarez-Gaumé, L., Bost, J.-B., Moore, G., Nelson, P., Vafa, C.: Bosonization on higher genus Riemann surfaces. Commun. Math. Phys. 112(3), 503–552 (1987)
Arnaudon M., Driver B.K., Thalmaier A.: Gradient estimates for positive harmonic functions by stochastic analysis. Stoch. Process. Appl. 117(2), 202–220 (2007)
Aubin, T.: Equations du type Monge–Ampère sur les variètès Kählériennes compactes. Bull. Sci. Math. (2) 102(1), 63–95 (1978)
Bardenet, R., Hardy, A.: Monte Carlo with Determinantal Point Processes. arXiv:1605.00361
Berman R.J.: Determinantal point processes and fermions on complex manifolds: large deviations and Bosonization. Commun. Math. Phys. 327(1), 1–47 (2014)
Berman R.J.: Kahler–Einstein metrics emerging from free fermions and statistical mechanics. J. High Energy Phys. (JHEP) 2011(10), 22 (2011)
Berman R.J.: A thermodynamical formalism for Monge–Ampere equations, Moser–Trudinger inequalities and Kahler–Einstein metrics. Adv. Math. 248, 1254 (2013)
Berman, R.J.: Statistical mechanics of permanents, real-Monge–Ampere equations and optimal transport. arXiv:1302.4045
Berman, R.J.: Kähler–Einstein metrics, canonical random point processes and birational geometry. In: AMS Proceedings of the 2015 summer research institute on algebraic geometry. arXiv:1307.3634 (2015, to appear)
Berman, R.J.: From Monge–Ampere equations to envelopes and geodesic rays in the zero temperature limit. arXiv:1307.3008
Berman, R.J.: On large deviations for Gibbs measures, mean energy and Gamma convergence. arXiv:1610.08219
Berman R.J., Boucksom S.: Growth of balls of holomorphic sections and energy at equilibrium. Invent. Math. 181(2), 337 (2010)
Berman R.J., Boucksom S., Witt Nyström D.: Fekete points and convergence towards equilibrium measures on complex manifolds. Acta Math. 207(1), 1–27 (2011)
Berman, R.J., Boucksom, S., Guedj, V., Zeriahi, A.: A variational approach to complex Monge–Ampere equations. Publ. math. de l’IHÉS, 1–67 (2012)
Berman, R.J., Onnheim, M.: Propagation of chaos, Wasserstein gradient flows and toric Kahler–Einstein metrics. arXiv:1501.07820
Berman, R.J., Darvas, T., Lu, C.H.: Convexity of the extended K-energy and the large time behaviour of the weak Calabi flow. Geom. Topol. (to appear). arXiv:1510.01260
Ben Arous G., Guionnet A.: Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Relat. Fields 108(4), 517–542 (1997)
Ben Arous G., Zeitouni O.: Large deviations from the circular law. ESAIM Probab. Stat. 2, 123–134 (1998)
Bloom T., Levenberg N.: Pluripotential energy and large deviation. Indiana Univ. Math. J. 62(2), 523–550 (2013)
Boucksom S., Essidieux P., Guedj V., Guedj V.: Monge–Ampere equations in big cohomology classes. Acta Math. 205(2), 199–262 (2010)
Borzellino J.E., Zhu S.-H.: The splitting theorem for orbifolds. Ill. J. Math. 38(4), 679–691 (1994)
Bodineau T., Guionnet A.: About the stationary states of vortex systems. Ann. Inst. Henri Poincaré Probab. Stat. 35, 205–237 (1999)
Braides A.: Γ-Convergence for Beginners. Oxford University Press, Oxford (2002)
Brézis H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)
Brøndsted A., Rockafellar R.T.: On the subdifferentiability of convex functions. Proc. Am. Math. Soc. 16, 605–611 (1965)
Caglioti E., Lions P.-L., Marchioro C., Pulvirenti M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143(3), 501–525 (1992)
Caillol J.M.: Exact results for a two-dimensional one-component plasma on a sphere. J. Phys. 42(12), L-245–L-247 (1981)
Chafaï D., Gozlan N., Zitt P.-A.: First-order global asymptotics for confined particles with singular pair repulsion. Ann. Appl. Probab. 24(6), 2371–2413 (2014)
Cheng S.Y., Yau S.T.: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28(3), 333–354 (1975)
Dembo A., Zeitouni O.: Large Deviation Techniques and Applications. Corrected Reprint of the Second (1998) Edition. Stochastic Modelling and Applied Probability, 38. Springer, Berlin (2010)
Doran, C., Headrick, M., Herzog, C.P., Kantor, J.: Numerical Kaehler–Einstein metric on the third del Pezzo. Commun. Math. Phys. 282(2), 357–393 (2008)
Donaldson, S.K.: Some numerical results in complex differential geometry. Pure Appl. Math. Q. 5(2) (2009), Special Issue: In Honor of Friedrich Herzebruch. Part 1, 571–618
Dupuis, P., Laschos, V., Ramanan, K.: Large deviations for empirical measures generated by Gibbs measures with singular energy functionals. arXiv:1511.06928
Ellis R.S., Haven K., Turkington B.: Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles. J. Stat. Phys. 101(5–6), 999–1064 (2000)
Eyink, G.L., Sreenivasan, K.R.: Onsager and the theory of hydrodynamic turbulence. Rev. Modern Phys. 78(1), 87–135 (2006)
Ferrari F., Klevtsov S., Zelditch S.: Random Kähler metrics. Nucl. Phys. B 869(1), 89–110 (2013)
Fine J.: Constant scalar curvature Kähler metrics on fibred complex surfaces. J. Differ. Geom. 68(3), 397–432 (2004)
Huichun, Z., Xiping, Z.: On a new definition of Ricci curvature on Alexandrov spaces. Acta Math. Sci. 30(6), 1949–1974 (2010)
Hough J.B., Krishnapur M., Peres Y.l., Virág B.: Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006)
Hultgren, J.: Permanental Point Processes on Real Tori, Theta Functions and Monge–Ampère Equation. arXiv:1604.05645
Kiessling, M.K.H.: Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure Appl. Math. 46, 27–56 (1993)
Kiessling, M.K.H.: Statistical mechanics approach to some problems in conformal geometry. Phys. A Stat. Mech. Appl. 279(1–4), 353–368 (2000)
Klevtsov, S.: Geometry and large N limits in Laughlin states. arXiv:1608.02928
Li P., Schoen R.: Lp and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math. 153(3-4), 279–301 (1984)
Messer J., Spohn H.: Statistical mechanics of the isothermal Lane–Emden equation. J. Stat. Phys. 29(3), 561–578 (1982)
Onsager L.: Statistical hydrodynamics. Supplemento al Nuovo Cimento 6, 279–287 (1949)
Schoen, R.: Lecture 1 and 2 in “nonlinear partial differential equations in differential geometry”. In: Hardt R., Wolf M. (eds.) IAS/Park City Math. Series. vol. 2 (1996)
Serfaty, S.: Coulomb gases and Ginzburg–Landau vortices. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2015)
Shiffman, B., Zelditch, S.: Distribution of zeros of random and quantum chaotic sections of positive line bundles. Commun. Math. Phys. 200(3), 661–683 (1999)
Sznitman, A.-S.: Topics in propagation of chaos. École d’Été de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Math., 1464, pp. 165–251. Springer, Berlin (1991)
Song Y., Tian G.: Canonical measures and Kähler–Ricci flow. J. Am. Math. Soc. 25(2), 303–353 (2012)
Tsuji, H.: Canonical measures and the dynamical systems of Bergman kernels. Preprint arXiv:0805.1829 (2008)
Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Amp‘ere equation. Commun. Pure Appl. Math. 31(3), 339–411 (1978)
Yau, S.T.: Non-linear analysis in geometry. Enseig. Math. (2) 33(1–2), 109–158 (1987)
Zeitouni O., Zelditch S.: Large deviations of empirical measures of zeros of random polynomials. Int. Math. Res. Not. IMRN 20, 3935–3992 (2010)
Zelditch, S.: Large deviations of empirical measures of zeros on Riemann surfaces. Int. Math. Res. Not. 2013(3), 592–664 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. -T. Yau
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Berman, R.J. Large Deviations for Gibbs Measures with Singular Hamiltonians and Emergence of Kähler–Einstein Metrics. Commun. Math. Phys. 354, 1133–1172 (2017). https://doi.org/10.1007/s00220-017-2926-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-017-2926-6